Multi lithology stratigraphic model under maximum erosion rate constraint

Description

Multi-lithology stratigraphic model under maximum erosion rate constraint R. Eymard‡ T. Gallouet D. Granjeon† R. Masson† Q.H. Tran† March 23, 2011 Abstract Non-linear single lithology or multi-lithology diffusion models have been widely used by sedimentologists and geomorphologists in the field of stratigraphic basin simulations to simulate the large scale depositional transport processes of sediments. Nevertheless, as noticed by many authors, erosion and sedimentation processes are non symmetric. Soil material must first be produced in situ by weathering processes prior to be transported by diffusion. This is usually taken into account through a prescribed maximum erosion rate of the sediments, but no mathematical description of the coupling with the diffusion model has been proposed so far. In this paper, we introduce a new mathematical formulation for the coupling of the weather limited erosion and the multi-lithology diffusion models, which appears as a non standard free boundary problem for a new variable acting as a limitor of the fluxes. One of the main advantages of this formulation, compared to existing discrete cou- pling models, is to enable the definition of efficient discretization schemes. A finite volume scheme with implicit time integration is introduced which is proved to be unconditionally stable in the l∞ norm for the sediment thickness, the sediment concentrations in the lithologies, and the flux limitor variables. A Newton algorithm with an iterative com- putation of the saturated constraints is used to solve efficiently the non-linear system resulting from the discretization.

• maximum erosion


• single lithology


• time integration


• rate constraint


• diffusion model


• multi-lithology diffusion


• diffusion


• dissymmetry between

Sujets

• Numerical ordinary differential equations
• Diffusion

Informations

• Publié par: pefav
• Nombre de lectures: 18
• Langue: English

Extrait

Multi-lithologystratigraphicmodelundermaximumerosion rateconstraint R.Eymard ‡ T.Galloue¨t § D.Granjeon † R.Masson † Q.H.Tran † March23,2011 Abstract Non-linearsinglelithologyormulti-lithologydiffusionmodelshavebeenwidelyused bysedimentologistsandgeomorphologistsinthefieldofstratigraphicbasinsimulations tosimulatethelargescaledepositionaltransportprocessesofsediments.Nevertheless,as noticedbymanyauthors,erosionandsedimentationprocessesarenonsymmetric.Soil materialmustfirstbeproducedinsitubyweatheringprocessespriortobetransportedby diffusion.Thisisusuallytakenintoaccountthroughaprescribedmaximumerosionrate ofthesediments,butnomathematicaldescriptionofthecouplingwiththediffusionmodel hasbeenproposedsofar.Inthispaper,weintroduceanewmathematicalformulation forthecouplingoftheweatherlimitederosionandthemulti-lithologydiffusionmodels, whichappearsasanonstandardfreeboundaryproblemforanewvariableactingasa limitorofthefluxes. Oneofthemainadvantagesofthisformulation,comparedtoexistingdiscretecou- plingmodels,istoenablethedefinitionofefficientdiscretizationschemes.Afinitevolume schemewithimplicittimeintegrationisintroducedwhichisprovedtobeunconditionally stableinthel ∞ normforthesedimentthickness,thesedimentconcentrationsinthe lithologies,andthefluxlimitorvariables.ANewtonalgorithmwithaniterativecom- putationofthesaturatedconstraintsisusedtosolveefficientlythenon-linearsystem resultingfromthediscretization.Theefficiencyofthemodelandthenumericalscheme isillustratedon2Dand3Dbasinsimulationexamples. 1Introduction Inrecentyears,therehasbeenagrowinginterestinthedevelopmentofmathematicaland numericalmodelsinstratigraphyandsedimentologyinresponsetotheneedforquantitative modelinginatraditionallyqualitativescience.Inthefieldofsequencestratigraphy,forward numericalmodelshaveshowntobeusefultoolstostudytheeffectsofeustasy,tectonics,and sedimentsupplyonfaciesdistributionandstratalgeometryofbasins. Oneofthemostimportantprocessinbasinevolutionistheerosionanddepositionmecha- nismofsediments.Authorsusuallydistinguishbetweenfluidflowanddynamic-slopemodels ofthesedimentationerosionprocesses(see[R92],[R97]).Thefirstonesusefluidflowequa- tionsandempiricalalgorithmstosimulatethetransportofsedimentsinthehydrodynamic flowfield(seee.g.[TH89]).Theyprovideanaccuratedescriptionofdepositionalprocesses forsmallscalesintimeandspace.Atlargerscalesuchasbasinscales,fluidflowmodelsare † InstitutFranc¸aisduPe´trole,1et4av.deBoisPre´au92852RueilMalmaisonCedex ‡ De´partementdeMathe´matiques,Universite´deMarnelaValle´e,5boulevardDescartes,ChampssurMarne, F-77454,Marneslavalle´e,Cedex2. § LATP,Universite´deProvence,39rueFre´de´ricJoliotCurie,13453Marseillecedex13. 1

computationallytooexpensiveanddynamic-slopeapproachesareusuallypreferred. Theselattermodelsactatamoremacroscopicscaleandusemassconservationequations ofsedimentscombinedwithdiffusivetransportlawsthataverageoverseveralprocesses(such asrivertransport,creep,slumpsandsmallslides).Theyprovideagooddescriptionof depositionalprocessesfortimescaleslargerthan,say,10 5 yrandbasinspacescales(see [AH89],[R92],[TS94],[R97],[G97],[GJD98],[GJ99]). Inmulti-lithologyapproaches,sedimentsaremodeledasamixtureofseverallithologies characterizedbydifferentgrainsizepopulations.Thesetofequationsaccountsforthemass conservationofeachlithologyinthebasin,knowingthesurficialfluxes,averticalcompaction model(usuallygivenbydepth-porosityempiricallawsforthelithologies),andanisostatic modelforthelithosphereflexure(usuallytakenasabeamequation). Themainsurficialtransportprocessisamulti-lithologydiffusionmodelintroducedby [R92]forwhichfluxesareproportionaltotheslopeofthetopographyaswellasalithology fractiondefinedatthesurfaceofbasin(seealso[G97],[GJ99],[QAD00]).Diffusioncoeffi- cientsarenon-linearfunctionsoftheelevation(orequivalentlythebathymetry)tomodelthe transitionfromnon-marinetomarinediffusionthatcandifferfromuptooneortwoorderof magnitude. However,itiswellknownthatsedimentationanderosionarenonsymmetricprocesses.To betransportedbysurficialprocesses,asdescribedby[AH89],materialmustfirstbeproduced insitubyweatheringprocessesdependingonclimate,elevation,compactionofsediments,... Thisproductionismodeledin[AH89]byaweatheringratealsocalledsoilproductionrate dependingonthesedimentdepthwithparametersfunctionofclimate.Then,oneachcellof thediscretemodel,thesedimentfluxesattheedgesareconstrainedsuchthaterosioncannot exceedtheavailablesoilthickness. In[G97]or[GJ99],themodelisslightlydifferentinthesensethatitdirectlyprescribes amaximumerosionrate(definedasthepartialtimederivativeofthesedimentthickness) dependingonclimate,elevation,composition,andtheageofthesediment. Asillustratedby[AH89],iftheweatheringrateislow,itcanbecomethedominantpro- cessforerosionwhichisnolongergovernedbyadiffusiontransportwhilesedimentationis stilldiffusiontransportlimited. Itappearsthatthecouplingoftheweatherlimitedmodelandthediffusiontransport modelisanessentialissueformodelingdepositionalprocesses.Nevertheless,thisquestionis notclearlyaddressedin[AH89]andisonlyreferredtoas“adiffusivetransportwithfluxes limitedbyavailablesediment”withoutadetaileddescriptionofthecouplingbetweenboth processes.Sofar,wearenotawareofanymathematicaldescriptionofthecouplingbetween weatheringlimitedanddiffusivetransportmodels.Themainobjectiveofthispaperisto proposesuchamathematicalmodelfromwhichweareabletoderivemoreefficientnumerical algorithmsforstratigraphicsimulations. Amodeltakingintoaccountthedissymmetrybetweenerosionandsedimentationisalso proposedin[R97].Theauthorusesacorrectionofthediffusioncoefficientbyaporosityratio (sedimentporosityoverthedepositionalporosity)equaltooneforsedimentationandlower thanoneforerosion.Thisratioactsasafluxlimitor,but,beingindependentonthefluxes (ortheslope),itwillfailtocorrectlyfreezetheerosionatthemaximumsoilproductionrate. Similarlyapointwisedependencyofthediffusioncoefficientontheerosionratewillalsofail since,asweshallseeinthesubsequentdevelopment,thefluxlimitorsatisfyingtheerosion 2